GEOMETRIC BOGOMOLOV CONJECTURE FOR ABELIAN VARIETIES AND SOME

26 Pages·2010·0.21 MB·English

GEOMETRIC BOGOMOLOV CONJECTURE FOR ABELIAN VARIETIES
AND SOME RESULTS FOR THOSE WITH SOME DEGENERATION
(WITH AN APPENDIX BY WALTER GUBLER: THE MINIMAL
DIMENSION OF A CANONICAL MEASURE)
KAZUHIKO YAMAKI
Introduction
Let K be a number field, or a function field of a curve over a base field k. We fix an
algebraic closure K of K. Let A be an abelian variety over K and let L be an ample line
bundle on A, and assume it is even, i.e., [−1]∗L = L. Then the canonical height function
ˆ
hL associat¡ed¢with L, also called the N´eron-Tate height, is a semi-positive definite quadratic
ˆ
form on A K . It is well-known that h (x) = 0 if x is a torsion point.
L
Let X be a closed subvariety of A. We put
n fl o
¡ ¢
fl
X(ϵ;L) := x ∈ X K flhˆ (x) ≤ ϵ
L
for a positive real number ϵ > 0. Then the Bogomolov conjecture for abelian varieties insists
that there should be ϵ > 0 such that X(ϵ;L) is not Zariski dense in X, unless X is a kind of
“exceptional” closed subvarieties, such as torsion subvarieties for example.
In the case where K is a number field, namely, in the arithmetic case, this conjecture was
solved more than ten years ago, known as a theorem of Zhang:
Theorem 0.1 (Corollary 3 of [21], arithmetic version of Bogomolov conjecture for abelian
varieties). Let K be a number field. If X is not a torsion subvariety, then there is ϵ > 0 such
that X(ϵ;L) is not Zariski dense in X.
The Bogomolov conjecture is originally a statement concerning the jacobian of a curve
and an embedding of the curve, that is, A is a jacobian and X is an embedded curve. This
is called the Bogomolov conjecture for curves, which is proved by Ullmo in [17] in case that
K is a number field at the same time when Zhang proved Theorem 0.1. The ideas of Ullmo
and Zhang are same — based on the equidistribution theory, which will be recalled in this
introduction.
When K is a finitely generated field over Q, a kind of arithmetic height functions can be
defined, after a choice of polarizations of K, due to Moriwaki [16]. It is still an arithmetic
setting namely, and the Bogomolov conjecture for abelian varieties with respect to the height
associated with a big polarization has been proved by Moriwaki himself. The classical
geometric height is also a kind of Moriwaki’s arithmetic height, but it does not arise from
a big polarization — rather a degenerate one. Hence we cannot say anything about the
geometric version of the conjecture with Moriwaki’s theory.
Date: July 4, 2010 (Version 4.2).
1
2 KAZUHIKO YAMAKI
How about the geometric case, namely the case where K is a function field over an
algebraically closed field k and the height is the classical geometric height? In this case,
we cannot expect the same statement as Theorem 0.1 because a subvariety defined over
the constant field can have dense small points. Accordingly, we have to reformulate the
conjecture, or have to consider it in a restricted situation.
The Bogomolov conjecture for curves is one of the problems studied for a long time. In
characteristic 0, Cinkir proved this conjecture in [7] recently. In positive characteristic, the
conjecture for curves is still open, but there are some partial answers such as in [15] by
Moriwaki and in [18, 19] by the author. In this case, the exceptional X’s are the isotrivial
curves. Another important result is the one due to Gubler. He proved in [9] the following
theorem:
Theorem 0.2 (Theorem 1.1 of [9]). Assume that there is a place v at which the abelian
variety A is totally degenerate. Then X(ϵ;L) is not Zariski dense in X for some ϵ > 0 unless
X is a torsion subvariety.
The exceptional X’s are the torsion subvarieties in this theorem as in the arithmetic case,
because there do not appear constant subvarieties in the totally degenerate case.
There are two things to do in this paper. One is to give the precise statement of our
geometric Bogomolov conjecture for abelian varieties, which is a reformulation of Zhang’s
theoremafterconsiderationofsubvarietiesdefinedoverk. Theotheristogivepartialanswers
to it for abelian varieties with some kind of degeneration.
We would like to give the statement of our conjecture now. Let G be the stabilizer of
X
X of a closed subvariety of an abelian variety A, and put B := A/G and Y := X/G . We
X X
call X a special subvariety of A if Y is the image of a closed subvariety of the K/k-trace of
B defined over k, up to the translation by a torsion point of B (cf. § 2.2). Note that if there
is a place v at which A is totally degenerate, then the notion of special subvarieties coincides
with that of torsion subvarieties since the K/k-trace is trivial. In §1, we will see that the
special subvariety has dense small points (cf. Corollary 2.8). Our geometric Bogomolov
conjecture insists that the converse should hold true:
Conjecture 0.3 (cf. Conjecture 2.9 and Remark 5.4). Let K be a function field. Let all A,
L and X be as above. Then there exists ϵ > 0 such that X(ϵ;L) is not Zariski dense in X
unless X is a special subvariety.
For an irreducible closed subvariety X ⊂ A and a place v of K, we can define an integer
b(X ), which we need in proposing our main result. Let us give rough description of it
v
although we refer to § 4.3 for the precise definition in terms of Raynaud extension. For
simplicity, we assume that A and X are defined over K and v is a place K. Further assume
that A has a model A over the ring of v-integers such that the reduction A˜ is a semi-abelian
variety. Then we have a surjective homomorphism q˜′ from A˜ to an abelian variety B over k
such that Kerq˜′ is an algebraic torus. Let X be the closure of X in A. Then the reduction
X˜ is a closed subset of A˜ and our b(X ) coincides with dimq˜′(X˜).
v
We can see that if there is a place v with dim(X/G ) > b((X/G ) ), then X is not a
X X v
special subvariety (cf. Proposition 5.1). Hence if our conjecture holds true, then such X
should not have dense small points. In fact, we will show the following result, which is our
main theorem of this paper:
GEOMETRIC BOGOMOLOV CONJECTURE 3
Theorem 0.4 (cf. Theorem 5.2 and Remark 5.4). Assume that there exists a place v such
that dim(X/G ) > b((X/G ) ). Then X(ϵ;L) is not Zariski dense for some ϵ > 0.
X X v
This theorem roughly says that a non-special subvariety of “relatively large” dimension in
some sense cannot have dense small points. By virtue of Theorem 0.4 together with Cinkir’s
theorem, we can see that the geometric Bogomolov conjecture almost holds for an abelian
variety A having a place v with b(A ) = 1 in characteristic 0. (cf. Theorem 5.3). Note that
v
the above theorem itself holds true in the case that K is a higher dimensional function field
as well as Theorem 0.2 (cf. Remark 5.4).
In the rest of this introduction, we would like to describe the idea of our proof. Before
that, let us recall the proof of Theorem 0.1 and that of Theorem 0.2, which gives us a basic
strategy.
First we recall the admissible metric. Let A be an abelian variety over C. Let L be an
even ample line bundle on X. It is well known that there is a canonical hermitian metric
h on L, called the canonical metric, such that [n]∗c (L,h ) = n2c (L,h ) and that
can 1 can 1 can
the curvature form c (L,h ) is smooth and positive. For a closed subvariety X ⊂ A of
1 can
dimension d, put
1
µ := c (L,h )d.
X,L 1 can
deg (X)
L
It has the total volume 1 and is smooth and positive on X.
Now let K be a number field. We recall what the equidistribution theorem says. Let X
be a closed subvariety of A. Let (x ) be a generic sequence of small points. Let σ be an
l l∈N
archimedean place, X the complex analytic space of X over σ, and let L be the restriction
σ σ
of L to X . Roughly speaking, the equidistribution theorem says that the Galois orbit of
σ
(x ) , approximatively as l → ∞, are equidistributed in X with respect to µ .
l l σ Xσ,Lσ
Let us recall the proof in the arithmetic case due to Ullmo and Zhang. It is done by
contradiction. Suppose we have a counterexample X for the Bogomolov conjecture. Then
takingthequotientifnecessary, wecaneasilyreduceourselvestothecasewherethestabilizer
is trivial and d := dimX > 0. Consider, for N ∈ N, a morphism
α : XN → AN−1, α(x ,...,x ) = (x −x ,...,x −x ).
1 N 2 1 N N−1
For large N, we can see that α gives a birational morphism XN → α(XN). We fix such
an N, writing X′ := XN and Y := α(X′) for simplicity. Then it induces an isomorphism
between some Zariski-dense open subsets U ⊂ X′ and V ⊂ Y. Let L′ and M be even ample
line bundles on X′ and Y respectively. Then we can see that X′ is again a counterexample
for the Bogomolov conjecture with respect to the line bundle L′. That implies that we can
find a generic sequence of small points (x ) , and we may assume they sit in U. Moreover,
l l∈N
we can see that the image (α(x )) is also a generic sequence of small points. By virtue
l l∈N
of the equidistribution theorem, (x ) and (α(x )) are equidistributed in X′ and Y with
l l∈N l l∈N
respect to µ and µ respectively, for an archimedean place σ. Furthermore since α
Xσ′,L′σ Yσ,Mσ
gives an isomorphism between U and V, we can conclude
µ | = α∗(µ | ).
Xσ′,L′σ U Yσ,Mσ V
Since both µ and α∗(µ ) are smooth forms, we have
Xσ′,L′σ Yσ,Mσ
∗
µ = α (µ )
Xσ′,L′σ Yσ,Mσ
4 KAZUHIKO YAMAKI
on X . The right-hand side however cannot be positive over the diagonal of X′ = XN. It is
σ
a contradiction since the left-hand side is positive.
HowaboutthecaseofGubler? Incontrasttothearithmeticcase,therearenoarchimedean
places in the geometric case. That fact had prevented us from enjoying an analogous proof
of the arithmetic case. To overcome that difficulty, Gubler used non-archimedean analytic
spaces over a non-archimedean place and their tropicalizations.
Let X ⊂ A be a closed subvariety of dimension d. To a place v of K, it is well-known that
the Berkovich spaces X ⊂ A can be associated. Gubler defined the canonical Chambert-
v v
Loir measure µ on X . Suppose here that A is totally degenerate. Then Gubler defined
Xv,Lv v v
the tropicalization Xtrop, which is denoted by val(X ) in his article, of X and showed that
v v v
it is a “d-dimensional polytope”. This plays the role of a counterpart of the complex space
over an archimedean place. Furthermore he investigated in detail the push-out µtrop to
Xv,Lv
the tropicalization of µ , describing it very concretely. In fact he showed that it is a
Xv,Lv
d-dimensional positive Lebesgue measure on the equi-d-dimesional polytope Xtrop.
v
Now the idea of Ullmo and Zhang can be applied to this situation. If there is a coun-
terexample to the Bogomolov conjecture, we can make the similar situation α : X′ → Y to
that of the arithmetic case, where X′ and Y are some closed subvarieties of abelian varieties.
Recall that X′ is also a counterexample of dimension d′ > 0 and that it has a generic net of
small points. We should note also that α is a generically finite morphism and the image of
the diagonal by α is one point. Tropicalizing them, we have
αtrop : (X′)trop → Ytrop,
v v
which is a morphism of polytopes. Since the subset corresponding to the diagonal contracts
to a point, there is a d′-dimensional face E such that F := αtrop(E) is a lower dimensional
face. Using the equidistribution theorem of himself to a generic net of small points, we can
obtain
αtrop(µtrop ) = µtrop
∗ Xv,L′v Yv,Mv
as well, where L′ and M respectively are even ample line bundles as before. It is impossible:
the left-hand side has a positive measure at a lower dimensional F, but the right one is the
d′-dimensional usual Lebesgue measure as mentioned. Thus a contradiction comes out.
Then how about the non-totally degenerate case? The basic strategy for the proof is
same—the equidistribution method. It is known that the canonical measure µ exists on
Xv,L
the Berkovich space X . Gubler defined in [11] the tropicalization Xtrop and studied the
v v
push-out µtrop of the canonical measure. He actually proved that Xtrop has the structure of
Xv,L v
a simplicial set and that µtrop can be described as
Xv,L
XN
(0.4.0) µtrop = r δ ,
Xv,L i ∆i
i=1
where ∆ runs through faces and δ is a usual relative Lebesgue measure on the simplex
i ∆i
∆ . On the other hand, he also proved in [10] that the equidistribution theorem holds true
i
in this situation. Thus we seem to have everything we need for the Bogomolov conjecture,
but we do not in fact. Then what we need more?
When we obtain the contradiction by using the equidistribution theorem, it was impor-
tant that the canonical form, or the canonical measure, is a “regular” one. If the canonical
GEOMETRIC BOGOMOLOV CONJECTURE 5
form was not smooth or positive in the arithmetic case, a contradiction would not come
out. In Gubler’s case also, it was the key that the tropicalization of the canonical measure
is the Lebesgue measure on the equi-d′-dimensional polytope. In the general case however,
lower dimensional ∆ ’s often appear in (0.4.0), and that is troublesome. In fact we can
i
make the‡same s·ituation as before, that is, we have a morphism αtrop : (X′)trop → Ytrop
v v
and αtrop µtrop = µtrop if we have a counterexample, but it is not sufficient to reach a
∗ Xv′,L′v Yv,Mv
contradiction because µtrop may contain a relative Lebesgue measure with a lower dimen-
Yv,Mv
sional support. That problem requires us more detailed analysis on the canonical measure.
We will show in the proof that µtrop does not have a component with lower dimensional
Yv,Mv
support than we expect.
This article is organized as follows. We will give some remarks on the trace of an abelian
variety in § 1. Those who are familiar with the trace will not have to read this section. In § 2,
we will formulate the geometric Bogomolov conjecture for abelian varieties. We will recall
in § 3 some results concerning our conjecture and will prove some immediate consequences.
We will describe in § 4 some basic properties on Berkovich spaces and their tropicalization
as far as we will need later. Our main results will be stated in § 5. In § 6, we will note some
properties on the canonical measures and will complete the proof of our main result. The
appendix is due to W. Gubler. In communicating with the author on the previous version
of this paper, he found a proof of the fact that the minimal dimension of the support of the
components of µtrop for ample L is exactly dimX −b(X ). Although we do not need this
Xv,L v
detailed information in the proof of our main theorem, it is quite interesting and will often
play an important role when you use the canonical measures.
Acknowledgments. The author would like to thank Prof. F. Kato, who gave lectures on
non-archimedean geometry to the author. The author also thanks Prof. W. Gubler for
giving a lot of valuable comments and suggestions, and for his appendix. This work has
been partially supported by KAKENHI(21740012).
Conventions and terminology. Throughout of this paper, let k be a fixed algebraically
closed field, and let K be the function field of a reduced irreducible smooth curve over k.
We fix an algebraic closure K.
Let K′/k be a field extension. For a scheme X over k, we write X := X × SpecK′.
K′ Speck
If ϕ : X → Y is a morphism of schemes over k, we write ϕ : X → Y for the base
K′ K′ K′
extension to K′.
When we say “height”, it means an absolute logarithmic height, for F = K with the
notation of [14, Chapter 3 §1].
For a finite extension K′ of K, let M denote the set of places of K′. If K′′ is a finite
K′
extension of K′, then there is a natural surjective map M → M . We put M :=
K′′ K′ K
lim M , where the K′ runs through all the finite extension of K in K, and call an element
←−K′ K′
of M a place of K. A place can be naturally regarded as a valuation of height 1. For a
K
place v ∈ M , let K be the completion of K with respect to v.
K v
6 KAZUHIKO YAMAKI
1. Descent of the base field of abelian varieties
We let L/k be any field extension, keeping our assumption k = k. We will discuss in this
section when an abelian variety over L can be defined over k, and will give remark on the
trace of an abelian variety.
We begin with a lemma:
Lemma 1.1. Let A and B be abelian varieties over L and k respectively. If ϕ : B → A is
L
an ´etale isogeny, then A and ϕ are defined over k: precisely, there exists a subgroup scheme
G of B over k such that Kerϕ = G⊗ L and hence A = (B/G) .
k L
Proof. Let N be the degree of ϕ. Let (B )[N] be the kernel of the N-times homomorphism.
L
Then Kerϕ ⊂ (B )[N] since Kerϕ is reduced by our assumption. Taking account that
L red
the field extension L/k is regular, we have
Kerϕ ⊂ (B )[N] = ((B[N]) ) = ((B[N]) ) ,
L red L red red L
which tells us that Kerϕ is defined over k, namely, there exists a subgroup scheme G of B
over k such that Kerϕ = G⊗ L.
k
We recall a quite fundamental theorem due to Chow here:
Theorem 1.2 (cf. II §1 Theorem 5 of [13]). Let A be an abelian variety over k and let B be
an abelian subvariety of A . Then there exists an abelian subvariety B′ ⊂ A with B′ = B.
L L
We can now show the following slight generalization of Theorem 1.2:
Proposition 1.3. Let A be an abelian variety over k and let G be a reduced closed subgroup
of A . Then there exists a closed subgroup G′ of A with (G′) = G.
L L
Proof. By Theorem 1.2, there exists an abelian subvariety G◦ ⊂ A such that G◦ is the
L
identity component of G. Consider the natural homomorphism ϕ′ : (A/G◦) → (A )/G. It
L L
is an ´etale isogeny since G is reduced, so by Lemma 1.1, there exist an abelian variety H
over k and a homomorphism ψ : A/G◦ → H such that ψ coincides with ϕ′. Now let G′ be
L
the kernel of the composition A → A/G◦ → H. Then we immediately find G′ = G.
L
Let B be an abelian variety over k and let ϕ : B → A be a smooth homomorphism
L
between abelian varieties over L. Then, as a corollary of Proposition 1.3, we can take an
abelian variety A′ over k and a homomorphism ϕ′ : B → A′ such that A = A′ and ϕ′ = ϕ.
L L
In fact, there exists a reduced closed subgroup G′ of B with Kerϕ = G′ by Proposition 1.3.
L
Then A′ := B/G′ suffices our requirement.
Next we will give remark on the Chow tr‡ace. Let F/·L be a field extension. Let A be an
abelian variety over L. Recall that a pair AF/k,TrF/k of an abelian variety AF/k over k
A
and a homomorphism TrF/k : (AF/k) → A × SpecF over F is called a F/k-trace, or
A F SpecL
Chow trace, if it satisfies the following universal property: for any abelian variety B over k
and for any homomorphism ϕ : B → A× SpecF, there exists a unique homomorphism
F Spec
L
ϕ′ : B → AF/k over k such that TrF/k◦ϕ′ = ϕ (cf. [13] and [14]).
A F
Lemma 1.4. TrF/k is finite and purely inseparable.
A
GEOMETRIC BOGOMOLOV CONJECTURE 7
Proof.‡By virtue·of Proposition 1.3, we can take a closed subgroup G′ ⊂ AF/k such that
G′ = KerTrF/k . Let π : AF/k → AF/k/G′ =: B be the quotient by G′. Then we have
F A
red
naturally a homomorphism ϕ : B → A× SpecF. By the universal property, we obtain
F Spec
L
the factorization ϕ′ : B → AF/k over k, and the‡universality·also says that ϕ′ ◦π = id .
AF/k
That concludes π is an isomorphism and hence KerTrF/k = 0, namely, TrF/k is finite
A A
red
and purely inseparable.
The uniqueness of the F/k-trace is immediate from the definition. We can find in [13] a
proof for the existence, but we should note one thing: in the definition of F/k-trace of [13,
VIII §8], Lang assumed that TrF/k is finite. This assumption is not necessary since it follows
A
from the definition automatically by virtue of Lemma 1.4 (cf. [14, the last line in p.138]).
Finally, we give remark on the homomorphism between the F/k-traces in char(k) = 0,
although it will not be needed in the sequel. Let A and B be abelian varieties over L and let
ϕ : A → B be a homomorphism. Then ϕ induces a unique homomorphism Tr(ϕ) : AF/k →
BF/k by the universal property.
Proposition 1.5 (char(k) = 0). Suppose that ϕ is surjective. Then Tr(ϕ) : AF/k → BF/k is
surjective.
Proof. Let us take an abelian subvariety A′ ⊂ A finite and surjective over B. Then we have
a composition of homomorphism (A′)F/k → AF/k → BF/k by the universality, and hence
we may assume that A is finite from the beginning, namely, ϕ is an isogeny. Let G be the
identity component of
¡ ¢
BF/k × (A× SpecF).
F (B×SpecLSpecF) SpecL ¡ ¢
Then it is an abelian variety over F and we have a natural homomorphism ψ : G → BF/k .
F
ˆ
It is also an isogeny, and its dual isogeny ψ is an ´etale isogeny since char(k) = 0. By virtue
of Lemma 1.1, we can take an isogeny ψ′ : G′ → BF/k such that ψ′ = ψ. Applying the
F
universality of the F/k-trace to the natural homomorphism G′ = G → A × SpecF,
F SpecL
we see that ψ′ factors as
G′ −−−→ AF/k −T−r−(ϕ→) BF/k.
Consequently, the induced homomorphism Tr(ϕ) is surjective since so is ψ.
2. Geometric Bogomolov conjecture
2.1. Small points. Let A be an abelian variety over K. For an even ample line bundle L
ˆ
on A, let us consider the canonical height function h . It is known to be a semi-positive
L
quadratic form on A(K). Let X be a closed subvariety of A. For each ϵ > 0, we put
n fl o
fl
X(ϵ;L) := x ∈ X(K)flhˆ (x) ≤ ϵ .
L
Lemma 2.1. Let A and X be as above. Let ϕ : A → B be a surjective homomorphism of
abelian varieties over K and put Y := ϕ(X). Let L and M be even ample line bundles on
A and B respectively. Then if X(ϵ;L) is Zariski-dense in X for any ϵ > 0, then Y(ϵ′;M) is
Zariski dense in Y for any ϵ′ > 0.
8 KAZUHIKO YAMAKI
Proof. Since L is ample, we can find a positive integer n such that L⊗n⊗ϕ∗(M)−1 is ample.
Then we have nhˆ ≥ ϕ∗hˆ , and hence
L M
‡n fl o·
fl
Y(ϵ;M) = ϕ x ∈ X(K)flhˆ (ϕ(x)) ≤ ϵ
M
‡n fl o·
fl
⊃ ϕ x ∈ X(K)flnhˆ (x) ≤ ϵ = ϕ(X(ϵ/n;L)).
L
The right-hand side is Zariski dense in Y by our assumption, and thus we have our assertion.
Let L and L be even ample line bundles on A. Then X(ϵ;L ) is Zariski dense for any
1 2 1
ϵ > 0 if and only if so is X(ϵ′;L ) for any ϵ′ > 0, by virtue of the above lemma. Accordingly,
2
the following definition makes sense:
Definition 2.2. We say that X has dense small points if X(ϵ;L) is Zariski dense in X(K)
for any ϵ > 0 and for some, hence any, even ample line bundle L on A.
We end this subsection with the following two basic lemmas on small points, which will
be used later:
Lemma 2.3. Let ϕ : A → B be a homomorphism of abelian varieties over K. Let X ⊂ A
be a closed subvariety and put Y := ϕ(X). Suppose that ϕ is a finite morphism.
(1) X has dense small points if Y has dense small points.
(2) Suppose further that ϕ is surjective. Then X has dense small points if and only if Y
has dense small points.
Proof. Let M be an even ample line bundle on B. Then L := ϕ∗M is also even and ample
and we have ϕ(X(ϵ;L)) = Y(ϵ;M). Then if X(ϵ;L) is not Zariski dense for any ϵ > 0, then
neither is not Y(ϵ;M) since ϕ is finite. This proves (1).
The assertion (2) follows immediately from (1) and Lemma 2.1.
Lemma 2.4. Let A and B be abelian varieties over K and let X ⊂ A and Y ⊂ B be closed
subvarieties. If X and Y has dense small points, then the closed subvariety X ×Y ⊂ A×B
has also dense small points.
Proof. Let p : A×B → B and q : A×B → B be the canonical projections. For even ample
line bundles L and M on A and B respectively, we write L£M := p∗L⊗q∗M. It is even
ample and we have hˆ = p∗hˆ +q∗hˆ . Accordingly we have
LˆM L M
(X ×Y)(2ϵ;L£M) ⊃ X(ϵ;L)×Y(ϵ;M),
and hence we obtain our assertion.
2.2. Special subvarieties and the conjecture. First of all, we wo‡uld like to d·efine the
notion of special subvarieties. For an abelian variety A over K, let AK/k,TrK/k denote
A
the K/k-trace of A. We refer to [14, Chapter 6] or 1 for its definition. Since AK/k is defined
¡ ¢
over k, we have the notion of k-points. We note AK/k(k) ⊂ AK/k K naturally.
Definition 2.5. Let A be an abelian variety over K.
GEOMETRIC BOGOMOLOV CONJECTURE 9
(1) Let X ⊂ A be an irreducible closed subvariety. Put B := A/G and Y := X/G ⊂
X X
B, where G is the stabilizer of X. We call X a special subvariety if there exist a
X
torsion point τ ∈ B(K) , and a closed subvariety Y′ ⊂ BK/k over k such that
tors
Y = TrK/k(Y′ )+τ.
B K
(2) A point σ ∈ A(K) is called a special point of A if the closed subvariety {σ} is a
special subvariety. We denote by A the set of special points of A.
sp
Let L be an even ample line bundle. Then we have
‡ · n fl o
fl
(2.5.1) A = A(K) +TrK/k AK/k(k) = x ∈ A(K)flhˆ (x) = 0 .
sp tors A L
In fact, the first equality is immediate from the definition. The second one follows from [14,
Theorem 4.5 and 5.4.2]. In particular, a special point is a point of height 0.
Lemma 2.6. Let ϕ : A → B be a surjective homomorphism of abelian varieties over K.
Then it induces a surjective homomorphism A → B .
sp sp
¡ ¡ ¢ ¢ ¡ ¢
Proof. The inclusion ϕ A K ⊂ B K is obvious. Moreover, the homomorphism ϕ
tors tors
inducesahomomorphismTrK/k(ϕ) : AK/k → BK/k (cf. 1). Nowitisclearthatϕ(A ) ⊂ B .
sp sp
Let us show the other inclusion. We can take an abelia¡n s¢ubvariety J ⊂ A such that ϕ|J
is a finite surjective homomorphism. Since a point x ∈ J K is of height 0 if and only if so
is ϕ(x). Therefore the induced map J → B is surjective by (2.5.1). Since J ⊂ A , we
sp sp sp sp
thus obtain our assertion.
Here is a remark. Suppose that ϕ is surjective. If chark = 0, we see that the induced
homomorphism TrK/k(ϕ) in the above proof is also surjective (cf. Proposition 1.5), but the
author does not know whether it holds in positive characteristic or not.
The following assertion says that the special subvarieties have dense small points:
Proposition 2.7. If X is a special subvariety of A, then X ∩A is dense in X.
sp
Proof. First let us consider the case where G = 0. Since our assertion is independent of
X
the translation by a torsion point, we may assume τ = 0 and hence TrK/k(Y′ ) = X for
A K
some Y′ ⊂ AK/k. Then our assertion is trivial since TrK/k(Y′(k)) ⊂ A and Y′(k) is dense
A sp
in Y′ .
K
Next let us consider the general case. Let ϕ : A → B := A/G be the quotient, and
X
put Y := X/G = ϕ(X). We have X = ϕ−1(Y) since G is the stabilizer of X. From the
X X
surjectivity of ϕ, we see
¡ ¡ ¢ ¢ ¡ ¢
(2.7.2) ϕ X K ∩A = Y K ∩B
sp sp
by¡Le¢mma 2.6. Since Y is a special subvariety of B and is stabilizer-fre¡e, w¡e s¢ee tha¢t
Y K ∩B isdenseinY asshownabove. By(2.7.2), wethusconcludethatϕ X K ∩A
sp ¡ ¢ sp
is dense in ϕ(X). On the other hand, take any x ∈ X K ∩ A with y = ϕ(x) for each
¡ ¡ ¢ ¢ sp
y ∈ ϕ X K ∩A . Since ϕ−1(Y) = X, we have
sp
x+(G ) ⊂ ϕ−1(y)∩A .
X sp sp
10 KAZUHIKO YAMAKI
Since x + (G ) is dense in x + G = ϕ−1(y), we find therefore ϕ−1(y) ∩ A is dense in
X sp X sp
ϕ−1(y). That says that the set of special points in the fiber of ϕ| : X → Y over y is also
X
den¡se¢in the fiber. Together with the fact that (2.7.2) is dense in Y, we can conclude that
X K ∩A is dense in X.
sp
In particular, we have the following:
Corollary 2.8. A special subvariety has dense small points.
Now let us propose the statement of our geometric Bogomolov conjecture for abelian
varieties, which insists that the converse of Corollary 2.8 should hold true:
Conjecture 2.9 (Geometric Bogomolov conjecture for abelian varieties (cf. Remark 5.4)).
X should not have dense small points unless it is a special subvariety.
We end this section with the following characterization of the special subvarieties.
Proposition 2.10. Let X ⊂ A be a closed subvariety and let G be the stabilizer of X. Put
X
B := A/G and Y := X/G . Then the following statements are equivalent to each other:
X X
(a) X is a special subvariety of A.
(b) Y is a special subvariety of B.
(c) There exist an abelian variety C over k, a homomorphism ϕ : C → B, a closed
K
subvariety Z′ ⊂ C, and a special point σ ∈ Y such that Y = ϕ(Z′ )+σ.
K
(d) There exist a variety W′ over k, a k-point w ∈ W′(k), a special point σ ∈ Y(K)
0
and a surjective morphism ψ : W′ → Y such that ψ(w ) = σ.
K 0
Proof. The equivalence between the first and the second statements is trivial from the defi-
nition. The implication from (b) to (c) and that from (c) to (d) are also trivial. Let us show
that (d) implies (b).
Let W′, w , σ and ψ be as in (d). For a fixed y ∈ B(K), we define T : B → B by
0 y
T (x) = x + y. First note that we can write σ = TrK/k(t) + τ with some t ∈ BK/k(k)
y B
and τ ∈ B(K) by (2.5.1). Then, by considering T (Y) and T ◦ ψ instead of Y and
tors −τ −τ
ψ respectively, we may assume that σ = TrK/k(t). Further, taking an alteration of W′ if
B
necessary, we may and do assume that W′ is nonsingular.
Let us consider the albanese morphism
α′ : W′ → Alb(W′)
w0
with respect to the base point w . Then
0
α := (α′ ) : W′ → Alb(W′) = Alb(W′ )
w0 w0 K K K K
is the albanese morphism of W′ with respect to w . By applying the universal property of
K 0
α to the morphism T ◦ψ, we obtain a homomorphism
w0 −σ
ϕ : Alb(W′ ) → B
K
with ϕ◦α = T ◦ψ. Then by the universal property of the K/k-trace, ϕ factors through
w0 −σ
the K/k-trace, that is, there is a homomorphism ϕ′ : Alb(W′) → BK/k such that
TrK/k◦(ϕ′ ) = ϕ.
B K